The On-Line Encyclopedia of Integer Sequences (OEIS)

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A319003

Number of ordered multiset partitions of integer partitions of n where the sequence of LCMs of the blocks is weakly increasing.
(history; published version)

#10 by Michael De Vlieger at Mon Jan 16 22:51:28 EST 2023

STATUS

proposed

approved

#9 by Andrew Howroyd at Mon Jan 16 22:26:02 EST 2023

STATUS

editing

proposed

#8 by Andrew Howroyd at Mon Jan 16 22:25:53 EST 2023

DATA

1, 1, 3, 7, 17, 38, 87, 191, 420, 908, 1954, 4160, 8816, 18549, 38851, 80965, 168077, 347566, 716443, 1472344, 3017866, 6170789, 12590805, 25640050, 52122784, 105791068, 214413852, 434007488, 877480395, 1772235212, 3575967030, 7209301989, 14523006820

OFFSET

1,2

0,3

LINKS

Andrew Howroyd, <a href="/A319003/b319003.txt">Table of n, a(n) for n = 0..50</a>

PROG

(PARI) seq(n)={my(M=Map()); for(m=1, n, forpart(p=m, my(k=lcm(Vec(p)), z); mapput(M, k, if(mapisdefined(M, k, &z), z, 1 + O(x*x^n)) - x^m))); Vec(1/vecprod(Mat(M)[, 2]))} \\ Andrew Howroyd, Jan 16 2023

KEYWORD

nonn,more

nonn

EXTENSIONS

a(0)=1 prepended and terms a(11) and beyond from Andrew Howroyd, Jan 16 2023

STATUS

approved

editing

#7 by Susanna Cuyler at Tue Sep 11 17:01:31 EDT 2018

STATUS

proposed

approved

#6 by Gus Wiseman at Mon Sep 10 19:14:05 EDT 2018

STATUS

editing

proposed

#5 by Gus Wiseman at Mon Sep 10 19:13:52 EDT 2018

COMMENTS

If we form a multiorder by treating integer partitions (a,...,z) as multiarrows LCM(a, ..., z) <= {z, ..., a}, then a(n) is the number of triangles of weight n.

CROSSREFS

Cf. A000837, A007716, A063834, A255397, A269134, A276024, A290103, A316222, A317545, A317546, A319001, A319004.

#4 by Gus Wiseman at Sun Sep 09 03:52:42 EDT 2018

NAME

Number of ordered multiset partitions of integer partitions of n where the sequence of LCMs of the blocks is weakly increasing.

#3 by Gus Wiseman at Fri Sep 07 07:52:39 EDT 2018

CROSSREFS

Cf. A000837, A007716, A063834, A255397, A269134, A276024, A290103, A316222, A317545, A317546, A319001, A319004.

#2 by Gus Wiseman at Fri Sep 07 07:39:04 EDT 2018

NAME

allocated for Gus WisemanNumber of ordered multiset partitions integer partitions of n where the sequence of LCMs of the blocks is weakly increasing.

DATA

1, 3, 7, 17, 38, 87, 191, 420, 908, 1954

OFFSET

1,2

COMMENTS

If we form a multiorder by treating integer partitions (a,...,z) as multiarrows LCM(a, ..., z) <= {z, ..., a}, then a(n) is the number of triangles of weight n.

EXAMPLE

The a(4) = 17 ordered multiset partitions:

{{4}} {{1,3}} {{2,2}} {{1,1,2}} {{1,1,1,1}}

{{1},{3}} {{2},{2}} {{1},{1,2}} {{1},{1,1,1}}

{{1,1},{2}} {{1,1,1},{1}}

{{1},{1},{2}} {{1,1},{1,1}}

{{1},{1},{1,1}}

{{1},{1,1},{1}}

{{1,1},{1},{1}}

{{1},{1},{1},{1}}

CROSSREFS

Cf. A000837, A007716, A063834, A255397, A269134, A276024, A290103, A316222, A317545, A317546, A319001.

KEYWORD

allocated

nonn,more

AUTHOR

Gus Wiseman, Sep 07 2018

STATUS

approved

editing

#1 by Gus Wiseman at Fri Sep 07 07:39:04 EDT 2018

NAME

allocated for Gus Wiseman

KEYWORD

allocated

STATUS

approved